where \((z,\alpha ) \in \mathbb {R}^n \times \mathbb {R}\) , \(0 < \varepsilon \ll 1\)  and the right-hand-side is \(C^{r \ge 1}-\) smooth in \((z,\varepsilon )\) . System (REF ) is in the standard form for fast-slow systems, with \(n\)  fast variables \(z\)  and a single slow variable \(\alpha \) . For the systems considered here, however, we do not require the existence of a critical manifold \(\lbrace (z,\alpha ) \in \mathbb {R}^{n+1} : f(z,\alpha )=0\rbrace \) . Thus (REF ) is not necessarily slow-fast in the geometric sense of [1]}, see also [2]}. Nevertheless, we shall refer to the limiting system
\(\begin{split}z^{\prime } &= \tilde{f}(z,\alpha ,0) , \\\alpha ^{\prime } &= 0 ,\end{split}\) 
